Not necessarily. We are certainly uncertain about random things, but we are also uncertain about some non-random things. Both can be represented as a distribution from which we can draw samples. So the mere act of drawing from a distribution does not imply randomness.
A good example is a pseudorandom number generator. There is nothing actually random about it. But we are uncertain of its next value, so we can describe it using a distribution and draw samples from it.
Isn't that the same in frequentist thinking?